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When steady supersonic flow hits a slim wedge, there may appear an oblique transonic shock attached to the vertex of the wedge, if the downstream pressure is rather large. This paper studies stability in certain weighted partial Hölder spaces of the oblique transonic shock attached to the vertex of a wedge, which is against steady supersonic flows, under perturbations of the upstream flow and the profile of the wedge. We show that under reasonable conditions on the upcoming supersonic flow and the slope of the wedge, such transonic shocks are structural stable. Mathematically, we solve an elliptic-hyperbolic mixed type in an unbounded domain, and the flow field is proved to be C1. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this per, we consider a special class of initial data for the three-dimensional incompressible Navier-Stokes equations with gravity. We show that, under such conditions, the incompressible Navier-Stokes equations with gravity are globally well posed, and the velocity minus gravity term has finite energy. The important features of the initial data is that the velocity fields minus gravity term are almost parallel to the corresponding vorticity fields in a very large space domain. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]